A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets

A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets Jos´ e Carlos R. Alcantud 1†, Roc´ ıo de Andr´ es Calle 1§ 1BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), Universidad de Salamanca, E-37007 Salamanca, Spain †E-mail: [email protected] §E-mail: r[email protected] Abstract We provide a methodology to perform an extensive and systematized analysis of problems where experts voice their opinions on the attributes of projects through a hesitant fuzzy decision matrix. This provides the decision-maker with ample information on which he or she can rely in order to make the final decision, in the form of segments instead of numbers. These segments derive from weighted average of new parametric expressions for two tenable indices of satisfaction, the distance to an ideal or the similarity to an anti-ideal, and permit to give a profuse unified picture of the relative performance of the projects. When the parameter grows, these indices tend to replicate the evaluation by respective simplistic expressions that only depend on the least, resp., the largest, evaluation and the number of evaluations in each cell. Keywords: Hesitant fuzzy set; Group decision making; Project evaluation; Segment-based evaluation. 1. Introduction The classical group decision making problem concerns the context where a group of experts have to make a decision on a set of alternatives, attending to either one or multiple criteria. The experts’ opinions about the alternatives are usually characterized by their knowledge or subjective ideas, which produces a rich environment of models in order to capture the setting and reach a final decision. The literature abounds with references about the decision making process under different positions8,9,13,24,31,36,49,53. It has long been recognized that fuzzy sets (FS) and fuzzy logic provide useful tools for the management of human subjectivity in decisionmaking contexts12,21,18,37. However in some practical problems, imprecise human knowledge (and especially group knowledge) cannot be suitably represented by fuzzy sets and some generalizations are needed. This was established as early as in Zadeh50. In this paper we are interested in a new, segment-based methodology that permits to perfom an extensive and systematized analysis of problems that are better modelled by Torra’s38 hesitant fuzzy sets (HFSs, originally considered by Grattan- Guinness16), which incorporate many-valued sets of memberships. The motivation for using this concept in decision making is clearly explained e.g., in Xu46. This reference justifies that hesitant fuzzy elements and sets have produced an extensive theoretical and applied literature 30,32,40,43,56. Furthermore, a recent authoritative survey of HFSs is Rodr´ ıguez et al.34. Here the authors summarize many useful and valuable decision making methods to solve hes- J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets itant fuzzy multi-criteria decision making problems and propose further applications of HFSs to decision making. 1.1. Our assumptions and research objectives We focus on the following common situation in multi-criteria decision analysis. We need to compare some alternatives or projects, and some experts evaluate their performance with respect to a set of attributes or characteristics. In this context the group knowledge on each project must be naturally represented by set-valued memberships, instead of just membership degrees as in fuzzy sets. Henceforth not only we permit imprecision or vagueness, but also a touch of uncertainty since we do not attach more value to a voiced opinion than to another one. Then the question arises: How do we analyze the problem of prioritizing these projects? The formal statement of this question refers to hesitant fuzzy decision matrices (HFDMs), i.e., matrices whose cells contain hesitant fuzzy elements (HFEs). These HFEs collect the opinions voiced by the experts on each attribute of the succesive projects. In our description rows are associated with projects and can be assimilated with HFSs. Thus we want to compare rows in these matrices on the basis of their relative performance (as alternatives or projects). The problem posed above, i.e., ranking HFSs or HFEs, has received attention from various authors recently. Xia and Xu44 and Farhadinia15 propose to use aggregating operators in order to associate a single HFE with each project. Then score functions give rankings of the aggregate HFEs. Xu and Xia47 rank the projects according to a direct appeal to distances. Finally, Zhou and Li54 design a lexicographic ranking that refines the Xia and Xu proposal44. A summary of these studies related to HFSs/HFEs ranking is given in Table 1. In order to make a broader analysis of these decision-making situations we draw inspiration from two sources. In the first place, we observe that the relative inadequacy of the projects (i.e., of their associated HFSs) can be estimated either by the ‘distance’ to an ideal HFS or the ‘similarity’ to an antiideal HFS, in the sense that the higher these evaluations the worse the project’s performance. Here we suggest respective novel parametric indicators for such proxies that incorporate the relative importance of the attributes through ex-ante allocations of weights. Their asymptotic behavior, i.e., the role of the parameter, is disclosed: when the parameter goes to infinity these indicators tend to provide an evaluation by respective simplistic expressions that only depend on the least, resp., the largest, evaluation and the number of evaluations on each attribute. In the second place, we draw inspiration from the Hurwicz approach to decision making under uncertainty26, which advocates for the combined use of ‘best and worst outcomes’ to assess the value of uncertain decisions. Thus the Hurwicz approach permits us to combine our two plausible parametric indices by their weighted sums, which includes both indices as extreme cases5. Their limit behavior replicates the case of the original indicators. Now for each project we obtain a segment instead of a single number, which can provide a richer analysis of the decision problem. Obviously, for any choice of the averaging aggregator a concrete ranking of projects arises. Table 1. Summary table of studies related to ranking of HFSs or HFEs Author(s) Tool(s)/method(s) Xia and Xu44 Aggregating operators and score functions Farhadinia15 Xu and Xia47 Distances and similarities Zhou and Li54 Lexicographic procedure We also report on the results of an experimental example that illustrates our proposal. In particular, we carry out a sensitivity analysis that permits to visualize the limit behaviour of our indexes in the analysis of problems characterized by HFDMs (e.g., hierarchization of projects characterized by HFSs). Finally, our approach is compared with other evaluation methods proposed by Xu46. J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets 1.2. Literature review: Project evaluation problems Multi-criteria decision making methods (MCDM) focus on the implementation of Decision Theory in real-life problems. One of the the most complex real situations is the evaluation of projects because it includes various factors and criteria. There exist different MCDM techniques to provide solutions to this problem. The appropriateness of the method depends on the specific decision situation39. Some examples of such decision contexts are: new product development projects10, energy projects19, information technology projects3, investment projects2, etc. Other contributions include some simple examples of different MCDM methodologies about project evaluation20,23,35. When the projects can begin on different time moments, the research on this problem is limited6,25,29. From another point of view, Fuzzy Set Theory has been extensively used to model uncertainty and vagueness associated with project information sources. And it has been gradually gaining importance as a tool in project selection4,7,17,27. Two milestones in this regard are Wang and Hwang41 (who develop a fuzzy integer programming model to gain an optimal investment portfolio), Chiu et al.11 and Wang et al.42 (who apply the fuzzy concept to the project selection process with the fuzzy multicriteria decision-making model (FMCDM) to select the optimal alternative). Within the extended field of Hesitant Fuzzy Sets, which allows for cases with several degrees of membership, there are many application papers that contribute to Multicriteria Decision Making Theory 1,22,44,48,52,51,54. Multiexpert multicriteria decision making under this requirement has been explored by Xia et al.45. A summary of these studies related to the evaluation problem is given in Table A.5. 1.3. Organization of the paper The remaining of this paper is organized as follows. Section 2 establishes some basic definitions. Section 3 introduces our proposals for ranking hesitant fuzzy sets, as well as results concerning the asymptotic behavior of our indices. In Section 4 we put in practice the methodology that permits to study the hierarchization of projects characterized by hesitant fuzzy sets. We visualize the asymptotic behavior of our indexes in a fully developed example, and then our results are confronted with the evaluations in existing approaches. We conclude in Section 5. 2. Notation and definitions For any (possibly infinite) set A,P∗(A)denotes the set of non-empty subsets of A, and F∗(A)denotes the set of non-empty finite subsets of A. Definition 1. 44A hesitant fuzzy element (HFE) is a non-empty, finite subset of [0,1]. The set of HFEs is denoted by F∗([0,1]). Henceforth we refer to X, a fixed set of alternatives. Definition 2. 38 Ahesitant fuzzy set (HFS) on Xis a function from Xto P∗([0,1]). A typical hesitant fuzzy set on Xis a function from Xto F∗([0,1]). HFS(X) means the set of HFSs on X, and the set of typical HFSs on Xis denoted by HFS(X). Unless otherwise stated, HFSs are assumed to be typical. Formally speaking, a (typical) HFS is a subset M⊆X×F∗([0,1]) such that for each x∈X, there is exactly one element hM(x)∈F∗([0,1]) such that (x,hM(x)) ∈M. Each HFS on Xdefines a set of membership values for each element of X, and in the case that the HFS is typical such set is always finite. HFEs represent the set of possible membership values of a typical hesitant fuzzy set at an alternative. By restricting ourselves to either F∗([0,1]) or P∗([0,1]), i.e., non-empty HFEs, we disregard ‘nonsense elements’ in each HFS: on each alternative, at least one assessment must be made. From a practical point of view, Xia and Xu44 show that the hesitant fuzzy set Mcan be represented as M={(x,hM(x)) |x∈X}. For example, following Torra38 we define M∗={(x,1)|x∈X} as the ideal or full HFS on X, and M−={(x,0)|x∈X} J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets as the anti-ideal or empty HFS on X. Clearly, when all HFEs involved in the definition of an HFS on Xare singletons we can identify such HFS with a fuzzy set (FS) on X. That is to say, HFEs of the form M={(x,hM(x)) |x∈X,hM(x) = {Mx}} can be identified with the FS on Xwhose membership function is µM:X−→ [0,1] xµM(x) = Mx For each typical hesitant fuzzy set Mon X, we denote hM(x) = {h1 M(x), ..., hlM(x) M(x)} where indexes are chosen so that h1 M(x)< . . . < hlM(x) M(x). In particular, the cardinality of the HFE hM(x)is lM(x) = |hM(x)|. Observe that if the set of membership values at an element is not finite (i.e., if we refer to a non-typical HFS) then such arrangement in increasing order cannot be made in general. In any case, because hM(x)is a set, repetitions are excluded by definition. Now we proceed to formalize the general concepts of distance and similarity between HFSs. Definition 3. [Xu and Xia 47] A distance measure between HFSs on Xis a function d:HFS(X)× HFS(X)−→ [0,1]that satisfies the following properties: for every M,N∈HFS(X), 1. 0 ⩽d(M,N)⩽1; 2. d(M,N) = 0 if and only if M=N; 3. d(M,N) = d(N,M). Definition 4. 47A similarity measure between HFSs on Xis a function s:HFS(X)×HFS(X)−→ [0,1] that satisfies the following properties: for every M,N∈HFS(X), 1. 0 ⩽s(M,N)⩽1; 2. s(M,N) = 1 if and only if M=N; 3. s(M,N) = s(N,M). There are similitudes between the latter concepts. When dis a distance measure between HFSs on X, the expression s=1−ddefines a similarity measure between HFSs on X. Conversely, when s is a similarity measure between HFSs on X, the expression d=1−sdefines a distance measure between HFSs on X. Besides Xu and Xia47, Xu46 collects many other examples of distance functions between HFSs in the literature. 3. Ranking typical HFSs: the segment approach In this Section we consider the analysis of the following problem. There are malternatives or projects whose performance with regard to ncriteria or attributes is evaluated by a team of experts (in a range from 0 to 1). Each expert can be hesitant on the performance of the projects, therefore he or she can emit any finite number of evaluations to express his or her doubts. For each project, all evaluations by the experts on each criteria are collected into a set of values. This presumes anonymity of the experts: all opinions are equally considered in this process. Formally, this produces an HFS associated with the project: for each attribute, a finite set of values in [0,1]is given. We face a problem under complete uncertainty: the importance of each particular appraisal is totally unknown. The opinions of the experts can be captured by a hesitant fuzzy decision matrix (HFDM), i.e., an m×nmatrix whose cells contain HFEs, in such way that its rows trivially define HFSs (one for each project). Columns correspond to respective evaluations of the projects by fixed criteria. Suppose that we need to rank or prioritize the projects. The problem arises: How do we analyze the decision problem posed? J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets 3.1. Analysis of the problem: the segment approach Several contributions have dealt with the problem posed above. Xia and Xu44 start by using aggregating operators in order to associate an HFE with each project, and then use a score function to rank them. Farhadinia15 proposes a variation with a different score function. Xu and Xia47 proceed in a more direct way: they rank the projects according to their distance to the ideal HFS. Finally, Zhou and Li54 do not produce evaluations of projects but give a lexicographic ranking that refines the proposal44. Our proposal intends to make a richer analysis by segments instead of points: with each project we associate a segment rather than a position or a number. It has two sources of inspiration. Firstly, we draw inspiration from the approach in Xu and Xia47. In order to analyze the relative performance of the projects (or of the HFSs that characterize them) we build on two relevant indicators, namely the ‘distance’ to the ideal HFS and the ‘similarity’ to the anti-ideal HFS. Both seem tenable indices of fitness for an HFS although of course, many distance and similarity indices can be used in analogy with the many proposals of distances between HFSs in the literature. In order to avoid confusions here we develop the model with a single concrete specification, namely, Definition 5 below that slightly echoes the use of the generalized hesitant weighted distance47. We leave the details of possible variations to the interested reader, i.e., specifications that replace our indicators in Definition 5 by expressions inspired on (i) the generalized hesitant weighted Hausdorff distance or the generalized hybrid hesitant weighted distance47 –among other distances between HFSs– or (ii) the ideas of the closedly related paper Xu and Xia48. We assume that each of the attributes has associated a weight wisuch that w1+. . .+wn=1. Weights are indicative of the relative importance of the attributes, hence a zero weight would mean a dispensable criteria that can be omitted in the analysis. This means that we do not lose generality if we assume wi>0 for each ihenceforth. Definition 5. Given λ>0 and w= (w1,...,wn)with wi>0 for each iand w1+. . . +wn=1, the λadjusted hesitant weighted distance to the ideal HFS is defined as ∆λ,w ahw(M) = n ∑ i=1 wi lM(xi) lM(xi) ∑ j=1 (1−hj M(xi))λ!1 λ (1) for each M∈HFS(M)and the λ-generalized hesitant weighted similarity to the anti-ideal HFS is defined as Σλ,w ahw(M) = 1− n ∑ i=1 wi lM(xi) lM(xi) ∑ j=1 (hj M(xi))λ!1 λ (2) for each M∈HFS(M). Observe ∆λ,w ahw(M) = 0 if and only if M=M∗, and Σλ,w ahw(M) = 0 if and only if M=M∗. Therefore both indicators share the characteristic that the higher the evaluation of a project, the worse its performance. In the case of Xia and Xu47, only the analogue of the first indicator is used∗ . In fact a direct inspection shows that when λ=1, our Definition 5 coincides with Xu and Xia’s general hesitant weighted distance47 between Mand M∗and therefore with their hesitant weighted Hamming distance between Mand M∗: Lemma 1. If λ=1 and w= (w1,...,wn)verifies wi>0 for each iand w1+. . . +wn=1, then ∆λ,w ahw(M) = Σλ,w ahw(M)for every M∈HFS(X). Proof. For every M∈HFS(M), ∆1,w ahw(M) = n ∑ i=1 wi lM(xi) lM(xi) ∑ j=1 (1−hj M(xi)) = = n ∑ i=1 wi lM(xi) lM(xi)− lM(xi) ∑ j=1 hj M(xi)!= = n ∑ i=1 wi− n ∑ i=1 wi lM(xi) lM(xi) ∑ j=1 hj M(xi)!= =1− n ∑ i=1 wi lM(xi) lM(xi) ∑ j=1 hj M(xi)!=Σ1,w ahw(M) ∗Distance and similarity measures under hesitant fuzzy environment and their properties were put forward in Xia and Xu47 J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets Secondly, we draw inspiration from the Hurwicz approach to decision making under uncertainty, which is very popular in Economics since its introduction in 1950 (cf., e.g., Luce and Raiffa26). In spirit it postulates the use of weighted sums of best and worst outcomes to assess the value of decisions. We can adapt it to the structure of our problem. In order to evaluate the acceptability of an HFS, both the ‘distance’ to the ideal HFS and the ‘similarity’ to the anti-ideal HFS are potentially useful. Instead of discarding one indicator in the benefit of the other, the segment approach permits us to combine both plausible indices. To be precise, in order to evaluate the hesitant fuzzy set Mwe define a value Λλ,w α(M) = α∆λ,w ahw(M)+(1−α)Σλ,w ahw(M)(3) which is a weighted sum of the distance to the ideal HFS and the similarity to the anti-ideal HFS. The weight α∈[0,1]can be conceived of as an index of ‘enviness’ because when α=1, the indicator coincides with ∆λ,w ahw, i.e., with the selected distance to the ideal HFS. When α=0, the indicator coincides with Σλ,w ahw, i.e., with the selected similarity to the anti-ideal HFS. Intermediate values permit to use the information in both indicators, and values close to 1, resp. 0, bias the indicator towards ∆λ,w ahw, resp. Σλ,w ahw. The higher the evaluation of an HFS by Λλ,w α, the worse its suitability. Therefore for each HFS we obtain a segment (as a function of α) instead of a single number, which can provide a more extensive analysis of the decision situation to the decision-maker. Obviously, for any fixed αa ranking of HFSs arises, although in general this ranking is dependent on the choice of the parameter. The decision maker can observe from a single drawing for which values of the parameter a given alternative is ranked first. Remark 1. As a consequence of Lemma 1, when λ=1 a unique ranking is obtained independently of the value of the parameter αbecause when w= (w1,...,wn)verifies wi>0 for each iand w1+. . . +wn=1, then Λ1,w α(M) = ∆1,w ahw(M) = Σ1,w ahw(M) for every M∈HFS(M). Fig. 1. The segment approach to the analysis of project evaluation problems. The use of the αand λparameters gives flexibility to our approach. Figure 1 graphically displays the structure and the flexibility of our approach. Besides the aforementioned intuition for the αparameter, subsection 3.2 below intends to help us understand the role of the λparameter. 3.2. Asymptotic behavior of the indicators: interpretations We proceed to check that using our indicators with ‘large’ values of the λparameter produces evaluations that are increasingly similar to those that derive from very simple indicators. Such indicators are crude evaluations that only rely on the number of different evaluations for each attribute and either the maximum or the minimum of such respective values. To this purpose let us define A(M) = ∑n i=1 wi lM(xi)maxj=1,...,lM(xi)(1−hj M(xi)) = =∑n i=1 wi lM(xi)(1−minj=1,...,lM(xi)hj M(xi)) for each M∈HFS(M)and B(M) = 1− n ∑ i=1 wi lM(xi)max j=1,...,lM(xi)hj M(xi) for each M∈HFS(M). Then our claim boils down to the following statement: Proposition 2. For every M ∈HFS(X), lim λ→∞ ∆λ,w ahw(M) = A(M) and also lim λ→∞ Σλ,w ahw(M) = B(M) Therefore, lim λ→∞ Λλ,w α(M) = αA(M)+(1−α)B(M) J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets for every M ∈HFS(X) and α∈[0,1]. Proof. We appeal to some basic properties of the lp norms on any Rt, defined as ||(x1,··· ,xt)||p= t ∑ j=1 |xj|p!1 p for every p⩾1.† We first observe that when M∈HFS(M), ∆λ,w ahw(M) = n ∑ i=1 wi lM(xi)||(1,lM(xi) ..........., 1)−(h1 M(xi),··· ,hlM(xi) M(xi))||λ Now it is easy to deduce the consequence limλ→∞∆λ,w ahw(M) = A(M): for each i=, ..., n, when λapproaches infinity the lλnorm on RlM(xi)approaches the l∞ or maximum norm defined as14 ||(x1,··· ,xt)||∞=max(|x1|,··· ,|xt|). The proof of the second claim is almost identical to the one above. The final statement can be trivially derived from the former ones. An intuitive interpretation is in order. ∆λ,w ahw(M)refers to similarity to an ideal HFS, and a proxy of that idea is given by the worst evaluation on each attribute, which is the information from which A(M)is designed. Similarly, Σλ,w ahw(M)refers to similarity to an anti-ideal HFS, and a proxy of that idea is given by the best evaluation on each attribute, which is the information on which B(M)is designed. 4. Experimental study In this section we give an experimental example to illustrate our proposal for the analysis of the hierarchization of projects respectively defined by hesitant fuzzy sets (HFSs). We also carry out a sensitivity analysis of the final outcomes in order to demonstrate the adaptability of the proposed model. Finally, we compare our conclusions with the evaluation methods proposed by Xu46, which provides experimental arguments supporting our approach. 4.1. Evaluation framework Our example builds on the discussion in Xu and Xia47 which is adapted from Kahraman and Kaya19. Accordingly, let us suppose a society which has to compare five energy projects, denoted by alternatives Ai(i=1,...,5). Four energy experts evaluate the performance of the five alternatives with respect to four main attributes or criteria (the example only collects all of the different possible values for each alternative and each attribute)‡ : •P1: Technological. In this criterion aspects like technical feasibility, technical risk, access to technology by local agents, maturity of projects, readiness of the local agents to implement the project, multiplicative effects on the local technology basis are taken into account. •P2: Environmental. Based on the project environmental impact. •P3: Socio-political. Included features like the consistency of the project with the society energy policy objectives, the political acceptance of the project, the social acceptance of the project, the scope of the project vs needs to be satisfied-urgency, the appropriateness of the implementing organization, etc. •P4: Economic. Estimated full cost of the project. The criteria significance fixed by the society is 15% for technological, 30% for environmental, 20% for socio-political and 35% for economic. Consequently the attribute weight vector used along the example is w= (0.15,0.3,0.2,0.35). The evaluations of the experts on the energy projects, which are based on the aforementioned criteria, are contained in a HFDM (see Figure 2 and Table 2). Fig. 2. Experimental study evaluation framework †When 0 <p<1 such expression does not define a norm, although ||(x1,··· ,xt)||p=∑tj=1|xj|pdoes (Maddox28). ‡For more details see Kahraman and Kaya19 and Xu and Xia47. J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets Table 2. Hesitant fuzzy decision matrix P1P2 A1{0.5,0.4,0.3} {0.9,0.8,0.7,0.1} A2{0.5,0.3} {0.9,0.7,0.6,0.5,0.2} A3{0.7,0.6} {0.9,0.6} A4{0.8,0.7,0.4,0.3} {0.7,0.4,0.2} A5{0.9,0.7,0.6,0.3,0.1} {0.8,0.7,0.6,0.4} P3P4 A1{0.5,0.4,0.2} {0.9,0.6,0.5,0.3} A2{0.8,0.6,0.5,0.1} {0.7,0.4,0.3} A3{0.7,0.5,0.3} {0.6,0.4} A4{0.8,0.1} {0.9,0.8,0.6} A5{0.9,0.8,0.7} {0.9,0.7,0.6,0.3} 4.2. Analysis of the hierarchization of projects: The segment approach In order to analyze the relative performance of the projects by means of the segment approach, we first need to produce the ‘distance’ to the ideal HFS and the ‘similarity’ to the anti-ideal HFS of each project, as measured by concrete realizations of λ in Definition 5. To be precise, we specify the outcomes when λ=1, λ=2 and λ=20. Finally, we illustrate the asymptotic behavior of the indicators when λis large enough by comparing these outcomes with the much simpler indicators in subsection 3.2. •Case λ=1. Table 3 shows the results of the computations for ∆λ,w ahw,Σλ,w ahw and Λλ,w α. As proven in Lemma 1, the evaluations when λ=1 are coincident hence the conclusion A5A3A4A1A2irrespective of which compromise index and value of αwe use. This consequence is shown in Figure 3 too. Table 3. Elements for the analysis when λ=1 Alternatives ∆λ,w ahw =Σλ,w ahw =Λλ,w α A10.477 A20.502 A30.402 A40.429 A50.355 Fig. 3. A graphical display of the indicators ∆1,w ahw =Σ1,w ahw =Λ1,w α •Cases λ=2,λ=20. Tables 4 and 5 show the results of the respective computations for these values. In order to compare the projects under a given choice of λ, the corresponding five segments Λλ,w αcan be drawn. This graphical analysis for the cases λ=2,λ=20 is performed in the respective Figures 4 and 5. Table 4. Elements for the analysis when λ=2 Index A1A2 ∆λ,w ahw 0.283 0.298 Σλ,w ahw 0.707 0.715 Λλ,w α0.707 −α0.424 0.715 −α0.417 Index A3A4 ∆λ,w ahw 0.286 0.287 Σλ,w ahw 0.581 0.635 Λλ,w α0.581 −0.295α0.635 −0.348α Index A5 ∆λ,w ahw 0.198 Σλ,w ahw 0.655 Λλ,w α0.655 −0.456α Table 5. Elements for the analysis when λ=20 Index A1A2 ∆λ,w ahw 0.217 0.227 Σλ,w ahw 0.795 0.786 Λλ,w α0.795 −0.577α0.786 −0.559α Index A3A4 ∆λ,w ahw 0.241 0.242 Σλ,w ahw 0.660 0.714 Λλ,w α0.660 −0.419α0.714 −0.471α Index A5 ∆λ,w ahw 0.153 Σλ,w ahw 0.773 Λλ,w α0.773 −0.620α •Case λ→∞. Table 6 shows the results of the computations for Aand Band also the respective values of Iαas a function of α∈[0,1]. Iα(Ai) = αA(Ai)+(1−α)B(Ai). These segments –one for each project– are uniquely determined by the HFDM. In Figure 6, the Iα(Ai)segments are plotted. We recall that they only depend on the least and J. C.R. Alcantud, R. de Andr´ es Calle / A segment-based approach to the analysis of project evaluation problems by hesitant fuzzy sets the largest evaluation and the number of evaluations in each cell. With respect to the asymptotic behavior it can be checked that the evaluations of the projects by the ∆λ,w ahw indicator are identical to the respective evaluations by Awhen λ=55, and the evaluations of the projects by the Σλ,w ahw indicator are identical to the respective evaluations by Bwhen λ=75 (with a 10−6precision). Table 6. Limit values of the indicators. Iαdenotes αA+(1−α)B. Index A1A2 A0.217 0.227 B0.795 0.786 Iα0.795 −0.578α0.786 −0.559α Index A3A4 A0.241 0.242 B0.660 0.715 Iα0.660 −0.419α0.715 −0.472α Index A5 A0.15325 B0.77425 Iα0.77425 −α0.621 A possible criticism to this approach is that it is fairly complex and certain factors (the λand αparameters) must be fixed. This seems to introduce ambiguity in the process of decisionmaking. Nevertheless we must point out that (i) this apparent inconvenience is common to many approaches in exactly the same setting, as subsection 4.3 below recaps; and (ii) the usual role of the analyst is to provide the decision-maker with as much information as possible, rather than making decisions. In this regard, note that our analysis provides visual information in the form of a two-dimensional graph for each choice of λ. The asymptotic behavior of these graphs (or the corresponding indexes) reveals that with only a few properly selected graphs, a complete assessment can be made. Fig. 4. A graphical display of the indicator Λ2,w α. Fig. 5. A graphical display of the indicator Λ20,w α. Fig. 6. A graphical display of the indicator Iα=αA+(1−α)B. 4.3. Discussion of the experimental study With the results of our experimental example set out, we now proceed to compare them with the rankings obtained from different methodologies that rank HFSs. We begin with the procedure in Xu and Xia47. Table A.1 contains rankings proposed by the generalized hesitant weighted distance (dghw), the generalized hesitant weighted Hausdorff distance (dghnh), the generalized hybrid hesitant weighted distance (dghhw) and the generalized hesitant ordered weighted distance (dghow). The authors give rankings for several choices of the λparameter that we adopt for comparison. In Xia and Xu44, Section 4, the authors proposed to use a GHFWAλoperator (generalized hesitant fuzzy weighted averaging operator, which requires to fix a weight vector and depends on a λfactor) in order to aggregate HFEs, and then rank the resulting HFEs according to their S1score S1(h) = h1+. . . +hlh lh . Rodr´ ıguez et al.34, Section 4, reported on many other alternative aggregators on HFEs, like GHFWGλ, GHFOWA or GHFOWG 44 or QHFOWA, HFMOWA and HFMOWG45. Furthermore, Farhadinia’s S2score or any other score on HFEs can be employed as an altermative to S1. Recall that Farhadinia15 proposed to start with a monotone non-decreasing sequence {δ(1),...,δ(n), . . .}of positive numbers and then use the score S2(h) = δ(1)h1+. . . +δ(lh)hlh δ(1)+. . . +δ(lh). In Table A.2 we have computed the prioritizations with the GHFWGλand GHFWAλaggregators, coupled with Xia and